Effective conductivity of two-dimensional heterostructures with non-screened potential

a28v322a

Effective Conductivity of Two-Dimensional Heterostructures with Non-Screened Potential

F. T. Vasko^1,2 and G.-Q. Hai¹

¹Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil

²Institute of Semiconductor Physics, NAS, 45 Pr. Nauky, Kiev, 03650, Ukraine

Received on 23 April, 2001

The effective magnetoconductivity tensor of selectively doped heterostructures with non-screened potential is evaluated for high-frequency regime. The lateral inhomogeneity of the donor distribution in the structure results in a large-scale variation of the energy levels. The contribution due to such inhomogeneity takes place in addition to the standard Drude mechanism of absorption and this mechanism is dominant for the high-frequency regime.

I Introduction

In high-frequency regime, if the photon frequency w or the cyclotron frequency w_c exceed the relaxation frequency (but corresponding energies are smaller than the mean electron energy), the linear response of electrons shows the Drude dispersion, i.e. the conductivity is found to be proportional to w^-2 or w_c^-2. This general dependency appears to be violated in semiconductor heterostructures with non-screened potential. The screening is not effective on a random potential due interface roughness or due to in-plane non-uniform d-doping in QW with two occupied subbands[1] or in multiple selectively doped QWs[2].

In the present paper, we describe the response of the 2D high-mobility electrons in selectively doped heterostructures taking into account an in-plane electron redistribution under non-screened random potential due to lateral inhomogeneity of d-doping (see Fig.1). Based on an electrostatic description of the non-screened variation of the potential, the effective conductivity tensor of the non-uniform electron gas is obtained from the linearized kinetic equation in Sec. II. The interplay between the Drude frequency dispersion, due to the standard relaxation processes, and the additional contribution, due to the planar large-scale inhomogeneity, is described in Sec. III. This frequency dependency of the effective conductivity is analyzed for the case of acoustic phonon scattering. Our numerical results demonstrate an essential modification on the response. The concluding remarks are presented in the last section.

II Effective Conductivity Tensor

The self-consistent description of the 2D electrons in selectively-doped structures with large-scale variations of donor distribution over the 2D plane is based on the Schrödinger and the Poisson equations. Within the first order approximation to the inhomogeneous potential w_x, the electron concentration n_xz can be written in the form n_xz = r_2D(e_F - w_x), where r_2D is the 2D density of states, e_F the Fermi energy, and y_z the wavefunction of the ground level. For the large-scale variation of the potential (under the condition l_c >> a_B, where l_c is the lateral correlation length of non-uniform donor distribution and a_{\scriptscriptstyle B} is the Borh radius), the solution of the Poisson equation is of a simple form[2]:

where dn_q is the Fourier component of the 2D donor concentration and Z is the average distance between the 2D electrons and the d-layer. Thus, the non-screened potential is suppressed if Z >> l_c. For further numerical estimations we assume a Gaussian correlation of variation of the donor concentration. Within the second order accuracy, the correlation function W(q) = òd(Dx)exp(-iq·Dx)áw_xw_x¢ñ is given by

where is the typical variation of the 2D concentration and n_2D = r_2De_F.

In a lateral electric field E_xexp(-iwt) and a transverse magnetic field H||OZ, the linear response of the 2D electrons within the dispersion law e_px = e_p+w_x (e_p = p²/2m is the kinetic energy and m is the effective mass) is described by the field-induced contribution to the distribution function: (f_eq+c_pq)exp(-iwt+iqx). Separating the symmetric and non-symmetric part of the distribution function, f_ex and c_pq, respectively, we write the linearized kinetic equation as follows (see [3] for details):

Here

_nel is the non-elastic collision integral, v = p/m, and { ¼} º dj( ¼) /(2p) stands for the average over the in-plane angle j. The collisionless evolution operator _q and the non-homogeneous term P_pq are given by

and

respectively, where n_e is the momentum relaxation frequency corresponding to the short range scattering mechanism, w_c=|e|H/(mc) the cyclotron frequency, E_q the Fourier component of the electric field, and e_Fx = e_F+w_x the non-uniform Fermi energy. The non-symmetric part of the distribution is determined by Eq. (4) and can be expressed through

The symmetric part of distribution is given by f_{e q} { ( q·v) c_pq} /w for the high-frequency limit.

The Fourier components of the induced charge r_q and current j_q are introduced through the standard relations: r_q = (2e²/L²)å_pf_eq, and j_q = (2e²/L²)å_pvc_pq. The average induced current is expressed through áf_{e q}ñ and áP_pqñ, where á¼ñ stands for the average over the random potential. Since the average values are proportional to d_q0, we obtain áj_qñ = d_q0j. After the summation over p we transform the induced current into

with

where the right side is written through the Fourier component of the concentration n_q = n_2D+r_2Dw_q. Separating the uniform and potential parts of electric field (under the condition curlE_x = 0), we write E_q as E_q = Ed_q0-iqF_q with E the homogeneous electric field. The random Fourier component of the potential F_q is determined from the Poisson equation with the high-frequency charge density r_q. Within the first order approximation to the random contribution, its solution appears to be proportional to (q·E)w_q. Substituting the random field into Eq. (5) and averaging according to Eq. (2), we obtain j through the effective conductivity tensor as j = E. The diagonal (non-diagonal) component of the conductivity () is written as

where s_d,^ corresponds to the uniform case. Since A_d(q) and A_^(q) depend on w and w_c, the above equation describes the frequency dispersion of the effective magnetoconductivity (including the case of cyclotron resonance). It also appears to be temperature dependent through the relaxation frequency n_e. A geneal expression for may be obtained after the evaluation of A_d,^(q) and the integration over q in Eq. (6).

III Spectral Dependencies

We consider now, in the absence of magnetic field, the modification on the Drude spectral dependency result from the second addendum to Eq. (6). Using an explicit expression for F_q and performing the average over non-screened random potential, we transform the non-uniform contribution as follows:

with

Further, performing the integral over q we find the effective conductivity in the form = (e²n_2D/mn_e)F(W,h) with the dimensionless function:

where W = w/n_e , h = 2l_F/l_c, g = 4Z/l_c, l = a_B/l_c, and V_W(hx) = .

The numerical calculation is performed for d-doped AlGaAs/GaAs heterojunction with /n_2D» 0.3. We have used » 76 ps and l_F » 14.7 mm in agreement with the experimental data[4] for the structures with high mobility. Figure 2 shows the frequency dependence of Re F(W,h) for different h and g. The parameter l is determined through h by l = (a_B/2l_F)h. The Drude dispersion for the ideal structure is given by the dotted curve in the figure. One can see that the random-induced absorption is comparable (or up to 10 times greater) to the Drude contribution and the hump-type dispersion takes place for the high-frequency region. The considered frequency corresponds to GHz frequency range.

IV Conclusion

In this work, we have studied the effective conductivity tensor of the selectively-doped heterostructures with non-screened large-scale random potential induced by the inhomogeneity of the doping layer. The presence of such a potential leads to a lateral variation of the energy levels and, consequently, modifies qualitatively the character of the frequency response of the system due to the contribution of non-uniform electric field. This new mechanism of the frequency dispersion is essential for the spectral region w

v_F/l_c and the obtained contribution is proportional to the mean square variation of the level in the weak disorder regime. Since the Drude absorption is suppressed as w^-2 at high frequency, the proposed mechanism appears to be a few times greater than the standard result (see Fig.2 ).

To conclude, the obtained spectral (or temperature) dependencies of effective conductivity will stimulate an experimental study of high-frequency response taking into account the lateral inhomogeneity of selectively-doped structures. Along with the frequency measurements of electron response, other characteristics of non-homogeneous structures, like 2D plasmon dispersion or cyclotron resonance, are also influenced by the non-screened variation of the levels. In addition, the temperature dependence of mobility and the static magnetotransport phenomena should be modified due to the analogous mechanism. In the static case we need to find f_eq taking into account the non-elastic collisions. These phenomena require a special consideration.

Acknowledgments

This work was supported by FAPESP and CNPq (Brazilian agencies).

References

[1]O.G. Balev, F.T. Vasko, F. Aristone, and N. Studart, Phys. Rev. B 62, 10212 (2000); F.T. Vasko, O.G. Balev and N. Studart, Phys. Rev. B 62, 12940 (2000).

[2]F.T. Vasko, and G.-Q. Hai, Phys. Rev. B, submitted.

[3]E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics, Pergamon Press, Oxford (1981).

[4]P.J. Burke, I.B. Spielman, J.P. Eisenstein, L.N. Pfeiffer, and K.W. West. Appl. Phys. Lett. 76, 745 (2000).

Publication Dates

• Publication in this collection
  26 Nov 2002
• Date of issue
  June 2002

History

• Received
  23 Apr 2001